A096711 -id:A096711 - cp's OEIS frontend

A096693 Balance index of each prime.

0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 4, 0, 0, 5, 1, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 1, 0, 1, 0, 1, 0, 2, 0, 2, 1, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Robert G. Wilson v, Jun 26 2004

nonn

a(n) = the number of values of k for which the n-th prime is equal to the arithmetic average of the k primes above and below it.

The average of the first n balance indexes appears to reach a global maximum of 0.588 when n = 85, (prime(85) = 439).

			a(3) = 1 because the third prime, 5, equals (3 + 7)/2.
a(16) = 3 because the sixteenth prime, 53, equals (47 + 59)/2 = (41 + 43 + 47 + 59 + 61 + 67)/6 = (31 + 37 + 41 + 43 + 47 + 59 + 61 + 67 + 71 + 73)/10.

C. H. Gribble, Table of n, a(n) for n=1,..., 10000.

Cf. A090403, A096695, A096705, A096706, A096707, A096708, A096709, A096711.

f[n_] := Block[{c = 0, k = 1, p = Prime[n], s = Plus @@ Table[ Prime[i], {i, n - 1, n + 1}]}, While[k != n - 1, If[s == (2k + 1)p, c++ ]; k++; s = s + Prime[n - k] + Prime[n + k]]; c]; Table[ f[n], {n, 105}]

b-file generator: {max_n = 10^4; for (n = 1, max_n, c = 0; k = 1; p = prime(n); s = p; while (k < n, s = s + prime(n - k) + prime(n + k); if (s == (2 * k + 1) * p, c++); k++;); print(n " " c);) ;}

Corrected and edited by Christopher Hunt Gribble, Apr 06 2010

A090403 Balanced primes: Primes which are both the arithmetic mean and median of a sequence of 2k+1 consecutive primes, for some k>0.

Original entry on oeis.org

5, 17, 29, 37, 53, 71, 79, 89, 137, 149, 151, 157, 173, 179, 193, 211, 227, 229, 257, 263, 281, 349, 353, 359, 373, 383, 397, 409, 419, 421, 433, 439, 487, 491, 563, 577, 593, 607, 631, 643, 653, 659, 677, 701, 709, 733, 751, 757, 787, 823, 827, 877, 947, 953

Offset: 1

Farideh Firoozbakht, Dec 07 2003

Union, for all k>0, of (2k+1)-balanced prime numbers, i.e., balanced prime of order k, which are primes p_n such that (2k+1)*p_n = Sum_{i=n-k..n+k} p_i, where p_i is the i-th prime.

			17 is in the sequence because 17 = (7 + 11 + 13 + 17 + 19 + 23 + 29)/7, (k = 3).
29 is in the sequence because 29 = (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59)/15, (k = 7).
37 is a member because 37 = (7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71)/17; 7 & 71 are eight primes away from 37.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A096693, A006562, A082077, A082078, A082079, A096697, A096698, A096699, A096700, A096701, A096702, A096703, A096704, A096711.

t[n_] := (For[k=1, !(SameQ[1/(2k+1)Sum[Prime[i], {i, n-k, n+k}], Prime[n]])&& k < n-1, k++ ];k);b[n_] := If[t[n]

is_A090403(p)={my(s=0,n); isprime(p) & for(k=1,-1+n=primepi(p),(s+=prime(n+k)+prime(n-k)-2*p)||return(1);s>p & return)} \\ M. F. Hasler, Oct 21 2012

Definition corrected by Franklin T. Adams-Watters, Apr 13 2006

Edited by M. F. Hasler, Oct 21 2012

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A096693 Balance index of each prime.

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